About twenty years ago Hawking made the remarkable suggestion that the black
hole evaporation process will inevitably lead to a fundamental loss of quantum
coherence. The mechanism by which the quantum radiation is emitted appears to
be insensitive to the detailed history of the black hole, and thus it seems
that most of the initial information is lost for an outside observer. However,
direct examination of Hawking's original derivation (or any later one) of the
black hole emission spectrum shows that one inevitably needs to make reference
to particle waves that have arbitrarily high frequency near the horizon as
measured in the reference frame of the in-falling matter. This exponential
red-shift effect associated with the black hole horizon leads to a breakdown of
the usual separation of length scales, and effectively works as a magnifying
glass that makes the consequences of the short distance, or rather, high energy
physics near the horizon visible at larger scales to an asymptotic observer.Comment: 6 pages, 2 figs. summary of lectures presented by Erik Verlinde at
  the 1994 Les Houches Summer School ``Fluctuating Geometries in Statistical
  Mechanics and Field Theory.'' (also available at http://xxx.lanl.gov/lh94/ )
  (based on work with Y. Kiem, K. Schoutens and H. Verlinde

Verlinde, Erik

English

arXiv

About twenty years ago Hawking made the remarkable suggestion that the black hole evaporation process will inevitably lead to a fundamental loss of quantum coherence. The mechanism by which the quantum radiation is emitted appears to be insensitive to the detailed history of the black hole, and thus it seems that most of the initial information is lost for an outside observer. However, direct examination of Hawking's original derivation (or any later one) of the black hole emission spectrum shows that one inevitably needs to make reference to particle waves that have arbitrarily high frequency near the horizon as measured in the reference frame of the in-falling matter. This exponential red-shift effect associated with the black hole horizon leads to a breakdown of the usual separation of length scales, and effectively works as a magnifying glass that makes the consequences of the short distance, or rather, high energy physics near the horizon visible at larger scales to an asymptotic observer.About twenty years ago Hawking made the remarkable suggestion that the black hole evaporation process will inevitably lead to a fundamental loss of quantum coherence. The mechanism by which the quantum radiation is emitted appears to be insensitive to the detailed history of the black hole, and thus it seems that most of the initial information is lost for an outside observer. However, direct examination of Hawking's original derivation (or any later one) of the black hole emission spectrum shows that one inevitably needs to make reference to particle waves that have arbitrarily high frequency near the horizon as measured in the reference frame of the in-falling matter. This exponential red-shift effect associated with the black hole horizon leads to a breakdown of the usual separation of length scales, and effectively works as a magnifying glass that makes the consequences of the short distance, or rather, high energy physics near the horizon visible at larger scales to an asymptotic observer.About twenty years ago Hawking made the remarkable suggestion that the black hole evaporation process will inevitably lead to a fundamental loss of quantum coherence. The mechanism by which the quantum radiation is emitted appears to be insensitive to the detailed history of the black hole, and thus it seems that most of the initial information is lost for an outside observer. However, direct examination of Hawking's original derivation (or any later one) of the black hole emission spectrum shows that one inevitably needs to make reference to particle waves that have arbitrarily high frequency near the horizon as measured in the reference frame of the in-falling matter. This exponential red-shift effect associated with the black hole horizon leads to a breakdown of the usual separation of length scales, and effectively works as a magnifying glass that makes the consequences of the short distance, or rather, high energy physics near the horizon visible at larger scales to an asymptotic observer

Verlinde, Erik P.

CERN Document Server

PostScript〉  processed by the SLAC/DESY Libraries on 20 Mar 1995.HEP-TH-9503120Black Hole Evaporation and Complementaritysummary of lectures presented by Erik VerlindeyAbout twenty years ago Hawking made the remarkable suggestion that the black holeevaporation process will inevitably lead to a fundamental loss of quantum coherence [1].The mechanism by which the quantum radiation is emitted appears to be insensitive to thedetailed history of the black hole, and thus it seems that most of the initial information islost for an outside observer. However, direct examination of Hawking's original derivation(or any later one) of the black hole emission spectrum shows that one inevitably needs tomake reference to particle waves that have arbitrarily high frequency near the horizon asmeasured in the reference frame of the in-falling matter. This exponential red-shift eectassociated with the black hole horizon leads to a breakdown of the usual separation oflength scales (see e.g. [2] and [3]), and eectively works as a magnifying glass that makesthe consequences of the short distance, or rather, high energy physics near the horizonvisible at larger scales to an asymptotic observer.Let us begin by reviewing the derivation of Hawking radiation in the s-wave sector,while at rst ignoring the eects due to gravitational back-reaction. We introduce twonull-coordinates u and v such that at a large distance r ! 1 we have u ! t + r andv! t  r. Following [1], we imagine sending a small test-particle backwards in time fromfuture null innity I+and letting it propagate all the way through to I (see gure 1.).To relate the form of this signal in the two asymptotic regions, we use the wave equationon the xed background geometry of the collapsing black hole.test waveI+I -v=v0singularityhorizonr=0Fig. 1based on work with Y. Kiem, K. Schoutens and H. Verlinde [4, 5]. This is a modied version of lecturenotes that will appear in the proceedings of the Trieste Spring School of 1993.yTH-Division, CERN, CH-1211Geneva 23, Switzerland and Institute for Theoretical Physics, Universityof Utrecht, P.O. BOX 80.006, 3508 TA Utrecht, the Netherlands1From the condition that the eld is regular at the origin r = 0 one deduces that theoutgoing s-wave outof the massless scalar eld  and the corresponding in-coming waveinare related by a reparametrizationin(v) = out(u(v)); (1)u(v) = v   4M log[(v0  v)=4M ]; (2)where M denotes the black hole mass and v0the critical in-going time, i.e. the location ofthe in-going light-ray that later will coincide with the black hole horizon. Thus an outgoings-wave with a given frequency ! translates to an in-signalei!u(v)= ei!v(v0  v4M) i4M!(v0  v); (3)that decomposes as a linear superposition of incoming waves with very dierent frequencies.The out-going modes b!(i.e. the Fourier coecients of out) are therefore related to thein-coming modes avia a non-trivial Bogoljubov transformation of the formb!=X!a+X!ay; by!=X!a+X!ay: (4)where !and !are given by the Fourier-coecients of (3), for example!= e i( !)v0e2M! (1   i4M!)2p!; (5)The transformation (4) is not invertible, and as a consequence pure in-states jini aremapped onto mixed out-states. In particular, the in-vacuum j0i evolves into the famousHawking state H, describing a constant ux of thermal radiation at the Hawking temper-ature TH=18M.In the above description the incoming particles described by in(v) with v > v0and theoutgoing particles represented by out(u) form independent sectors of the Hilbert space,because the corresponding eld operators commute with each other. The underlying clas-sical intuition is that the elds in(v) with v > v0will propagate into the region behindthe black hole horizon, and thus become unobservable from the out-side. However, here wehave ignored the fact that the in-falling particles in fact do interact with the out-going ra-diation, because they slightly change the black hole geometry. Furthermore, it can be seen2from (3) that a generic out-going wave carries a diverging stress-energy near the horizon.This indicates that gravitational interactions may indeed be important.Consider a spherical shell of matter with energy M that falls in to the black hole atsome late time v1. At this time the Schwarzschild radius slightly increases with an amount2M , and as a consequence the time v0at which the horizon forms will also change veryslightly (see gure 2). A simple calculation shows thatv0=  c Me (v1 v0)=4M; (6)where, c is a constant of order one. At rst it seems reasonable to ignore this eect as longas the change M is much smaller thanM , but, it turns out that, due to the diverging red-shift, the variation in v0, although very small, has an enormous eect on the wave-functionout(u) of an out-going particle.Shell00+ v0δv = vv = vNewOldHorizonEvent OutgoingParticleInfallingFig 2.By combining (1), (2) and (6) one easily veries that as a result of the in-falling shell, theoutgoing particle-wave is delayed by an amount that grows rapidly as a function of uout(u)! out(u  4M log(1  cM4Me(u v1)=4M)): (7)Notice that even for a very small perturbation M the argument of the eld outgoes toinnity after a nite time ulim u1  4M log(M=M). The physical interpretation of thisfact is that a matter-particle that is on its way to reach the asymptotic observer at some3time u > ulimwill, as a result of the additional in-falling shell, cross the event-horizon andbe trapped inside the black-hole horizon.To take the eect (7) into account in the quantum theory, we divide up the in-fallingmatter in a classical piece plus a small quantum part that is described in terms of a quantumeld in(v). Of course, v0is mainly determined by the classical in-falling matter, but inaddition there is a small contribution that depends on the elds in(v). For simplicity weassume here that the only eect is that the mass and thus the Schwarschild radius changesas in (6). In this way we ndzv0= vcl0  cZ1vcl0dv e(vcl0 v)=4MTin(v) (8)where Tin(v) denotes the stress-energy tensor of the in(v) with support v > vcl0.With this correction we now re-calculate the commutator of the outgoing eld out(u)for late times with the in-coming eld in(v) for v > vcl0. For this calculation we take thesame relation (1), that formed the starting point of Hawking's derivation, but we includein the reparametrization (2) the seemingly negligible quantum contribution in v0. Fromthe fact that the stress-tensor generates coordinate transformations we deduce[v0; in(v)] =  ic exp((vcl0  v)=4M)@vin(v); (9)or equivalentlyeiv0in(v)e iv0= in(v   4M log(1 c4Me(vcl0 v)=4M)); (10)Combining this result with (1) and (2) we obtain the following algebra for the in  andout-eldsout(u)in(v) = exp(ic e(u v)=4M@u@v)in(v)out(u); (11)which is valid at for v > vcl0. The `exchange algebra' (11) is the quantum implementationof the gravitational back-reaction (7) of the in-falling matter on the out-going radiation.Notice that this algebra is non-local, and, furthermore, that the non-locality grows expo-nentially with time.These gravitational interactions lead to a modication of the quantum state of theout-going radiation. To describe this modication we introduce addiditional modes c!zFor a detailed discussion of the angular dependence we refer to [5].4describing the in-falling particles at the horizon. In terms of these the original statecomputed by Hawking can be written as:j iHawking=1NexpnZ10d! e 4M!by!cy!oj0ib j0ic(12)But when one takes into account the correction (8) due to the gravitational backreactionone obtains for nal statej ifinal= U j iHawking(13)whereU = TexphiZ Zv0dudv e(u v)=4MTuu(u)Tvv(v)i(14)Notice again the interaction Hamiltonian wich appears in U contains a pre-factor thatgrows exponentially with time. This makes clear that these interaction become non-neglibleat practically the same time when the black hole radiation starts to appear.We want to emphasize that in the above derivation we did not make any assumptionsother than those already made in the usual derivation of Hawking evaporation. The onlyextra ingredient that we took into account is the small contribution to v0due to the eld. In this sense the relation (11) appears to be unavoidable and independent of whichscenario one happens to believe in. Therefore, it is clear that the presence of these largecommutators implies that the standard semi-classical picture of the black hole evaporationprocess needs to be drastically revised. In particular, it tells us that, due to the quantumuncertainty principle, we should be very careful in making simultaneous statements aboutthe in-falling and out-going elds.Our result supports the physical picture that there is a certain complementarity be-tween the physical realities as seen by an asymptotic observer and by an in-falling ob-server. Indeed, for the latter, the in-falling matter will simply propagate freely withoutany perturbation, but he (or she) will not see the out-going radiation. For the asymptoticobserver, on the other hand, the Hawking radiation is physically real, while the in-fallingmatter will appear to evaporate completely before it falls into the hole. Finally, we expectthat our result may also be of importance in relation with the entropy of black holes. In[6] it was noted that a naive free eld calculation of the one-loop correction to the blackhole entropy gives an innite answer. This innity arises due to the diverging contributionof states that are packed arbitrarily close to the horizon. Our result suggests a possibleremedy of this problem, because it shows that in the coordinate system appropriate forthis calculation the in- and out-going elds no longer commute when they come very closeto the horizon. We believe that this will eectively reduce the number of allowed states,and thereby eliminate the diverging contribution in the entropy calculation.5Acknowledgements.The research of E.V. is partly supported by an Alfred P. Sloan Fellowship, and by aFellowship of the Royal Dutch Academy of Sciences.References[1] S.W. Hawking, Commun. Math. Phys. 43 199 (1975) 199.[2] G. 't Hooft, Nucl. Phys.B335 (1990) 138, and \Unitarity of the Black Hole S-Matrix,"Utrecht preprint THU-93/04.[3] T. Jacobson, Phys. Rev. D48 (1993) 728[4] K. Schoutens, E. Verlinde and H. Verlinde, Black Holes and Quantum Gravity toappear in the Proceedings of the 1993 Trieste Spring School on Gauge Theory, Gravityand Strings, editors: R. Dijkgraaf etal .[5] Y. Kiem, E. Verlinde and H. Verlinde, Quantum Horizons and Complementarity,CERN/Princeton preprint CERN-TH.7469/94, PUPT-1504. hep-th/9502074[6] G. 't Hooft, Nucl. Phys. B 256 (1985), 727.6

Black Hole Evaporation and Complementarity

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